(sec^2x + csc^2x)(sec^2x - csc^2x) = tan^2x - cot^2x(sec2x+csc2x)(sec2x−csc2x)=tan2x−cot2x
(1/cos^2x + 1/sin^2x)(1/cos^2x- 1/sin^2x) = sin^2x/cos^2x - cos^2x/sin^2x(1cos2x+1sin2x)(1cos2x−1sin2x)=sin2xcos2x−cos2xsin2x
((sin^2x + cos^2x)/(cos^2xsin^2x))((sin^2x - cos^2x)/(cos^2xsin^2x)) = sin^2x/cos^2x - cos^2x/sin^2x(sin2x+cos2xcos2xsin2x)(sin2x−cos2xcos2xsin2x)=sin2xcos2x−cos2xsin2x
(sin^2x- cos^2x)/(cos^4xsin^4x) = (sin^4x - cos^4x)/(sin^2xcos^2x)sin2x−cos2xcos4xsin4x=sin4x−cos4xsin2xcos2x
(sin^2x -cos^2x)/(cos^4xsin^4x) = ((sin^2x+ cos^2x)(sin^2x - cos^2x)/(sin^2xcos^2x)sin2x−cos2xcos4xsin4x=((sin2x+cos2x)sin2x−cos2xsin2xcos2x
(sin^2x - cos^2x)/(cos^4xsin^4x) = (sin^2x - cos^2x)/(sin^2xcos^2x)sin2x−cos2xcos4xsin4x=sin2x−cos2xsin2xcos2x
As you can see, the two sides are unequal, so the identity is false.
Hopefully this helps!